SCHWEITZER 1981 Problem 10

Let $P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $p$ is zero outside the interval $[-1,1]$ and inside this interval it is between the positive numbers $c$ and $d$ ( $c < d$ ). Prove that there is no distribution whose convolution square equals $P$ .

T. F. Mori, G. J. Szekely